Trigonometry


  1. The value of sin² 65° + sin² 25° + cos² 35° + cos² 55° is









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    Expression = sin² 65° + sin² 25° + cos²35° + cos² 55°
    = sin² 65° + sin² (90° – 65°) + cos² 35° + cos² (90° – 35°)
    = sin² 65° + cos² 65° + cos² 35° + sin² 35°
    = 1 + 1 = 2

    Correct Option: C

    Expression = sin² 65° + sin² 25° + cos²35° + cos² 55°
    = sin² 65° + sin² (90° – 65°) + cos² 35° + cos² (90° – 35°)
    = sin² 65° + cos² 65° + cos² 35° + sin² 35°
    = 1 + 1 = 2


  1. If sin θ = 0.7, then cos q, 0 ≤ θ < 90°, is









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    sin θ = 0.7
    ∴ cos θ
    = √1 - sin² θ= √1 - (0.7)²
    = √1 - 0.49= √0.51

    Correct Option: C

    sin θ = 0.7
    ∴ cos θ
    = √1 - sin² θ= √1 - (0.7)²
    = √1 - 0.49= √0.51



  1. The value of q, which satisfies the equation tan²θ + 3 = 3 secθ, 0° ≤ θ < 90° is









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    tan²θ + 3 = 3 sec θ
    ⇒ sec²θ – 1 + 3 = 3 sec θ
    ⇒ sec²θ – 3 sec θ + 2 = 0
    ⇒ sec²θ – 2 sec θ – sec θ + 2 = 0
    ⇒ secθ (sec θ – 2) – 1 (sec θ – 2) = 0
    ⇒ (sec θ – 2) (sec θ – 1) = 0
    ⇒ secθ = 2 or 1
    ⇒ θ = 60° or 0°.

    Correct Option: D

    tan²θ + 3 = 3 sec θ
    ⇒ sec²θ – 1 + 3 = 3 sec θ
    ⇒ sec²θ – 3 sec θ + 2 = 0
    ⇒ sec²θ – 2 sec θ – sec θ + 2 = 0
    ⇒ secθ (sec θ – 2) – 1 (sec θ – 2) = 0
    ⇒ (sec θ – 2) (sec θ – 1) = 0
    ⇒ secθ = 2 or 1
    ⇒ θ = 60° or 0°.


  1. If (1 + sin A) (1 + sin B) (1 + sin C) = (1 – sin A) (1 – sin B) (1 – sin C), 0 < A, B, C < (π / 2) then each side is equal to









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    (1 + sin A) (1 + sin B) (1 + sin C) = (1 – sin A) . (1 – sin B) (1 – sin C) = x (Let)
    ∴ x . x = (1 + sin A) (1 + sin B) (1 + sin C) (1 – sin A) (1 – sin B) (1 – sin C)
    ⇒ x² = (1 – sin² A) (1 – sin² B) (1 – sin² C)
    ⇒ x² = cos²A . cos²B . cos²C
    ⇒x = ± cos A . cos B . cos C

    ∵ 0 < A, B, C <
    π
    2

    ∴ x = cos A . cos B . cos C

    Correct Option: B

    (1 + sin A) (1 + sin B) (1 + sin C) = (1 – sin A) . (1 – sin B) (1 – sin C) = x (Let)
    ∴ x . x = (1 + sin A) (1 + sin B) (1 + sin C) (1 – sin A) (1 – sin B) (1 – sin C)
    ⇒ x² = (1 – sin² A) (1 – sin² B) (1 – sin² C)
    ⇒ x² = cos²A . cos²B . cos²C
    ⇒x = ± cos A . cos B . cos C

    ∵ 0 < A, B, C <
    π
    2

    ∴ x = cos A . cos B . cos C



  1. The simplest value of cot 9° cot 27° cot 63° cot 81° is









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    Expression
    = cot 9°. cot 27°. cot 63° . cot 81°
    = cot 9°. cot 27°. cot (90° – 27°) . cot (90° – 9°)
    = cot 9° . cot 27° . tan 27° . tan 9° [tan (90° – θ)
    = cot θ; cot (90° – θ) = tan θ ]
    = cot 9° . tan 9° . cot 27° tan 27°
    = 1 [tan θ . cot θ = 1]

    Correct Option: B

    Expression
    = cot 9°. cot 27°. cot 63° . cot 81°
    = cot 9°. cot 27°. cot (90° – 27°) . cot (90° – 9°)
    = cot 9° . cot 27° . tan 27° . tan 9° [tan (90° – θ)
    = cot θ; cot (90° – θ) = tan θ ]
    = cot 9° . tan 9° . cot 27° tan 27°
    = 1 [tan θ . cot θ = 1]